AP Calculus BC Flashcards: Limits At Infinity And Horizontal Asymptotes

Study Limits At Infinity And Horizontal Asymptotes in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus BC

Limits At Infinity And Horizontal Asymptotes

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QUESTION

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Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$ .

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ANSWER

$y = 2$ . Fourth-degree terms dominate: $\frac{2}{1} = 2$ .

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What this deck covers

This deck focuses on Limits At Infinity And Horizontal Asymptotes, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$ .

Answer: $y = 2$ . Fourth-degree terms dominate: $\frac{2}{1} = 2$ .

Flashcard 2: What is the horizontal asymptote of $f(x) = \frac{2x^2 + x}{4x^2 + 3}$ ?

Answer: $y = \frac{1}{2}$ . Quadratic coefficients: $\frac{2}{4} = \frac{1}{2}$ .

Flashcard 3: State the horizontal asymptote for $f(x) = \frac{3x - 4}{x + 5}$ .

Answer: $y = 3$ . Linear terms give ratio $\frac{3}{1} = 3$ .

Flashcard 4: Identify the horizontal asymptote of $f(x) = \frac{x^2}{x^2 - x + 1}$ .

Answer: $y = 1$ . Same degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 5: What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 6: What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$ ?

Answer: $y = 0$ . Numerator degree less than denominator degree.

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 8: State the horizontal asymptote for $f(x) = \frac{x^2 + 2x}{x^2 - x}$ .

Answer: $y = 1$ . Equal-degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 9: Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$ .

Answer: $y = 0$ . Linear over cubic approaches 0.

Flashcard 10: Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$ .

Answer: $y = 0$ . Linear over cubic approaches 0.

Flashcard 11: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$ ?

Answer: $y = \frac{5}{2}$ . Ratio of leading coefficients for equal-degree polynomials.

Flashcard 12: Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$ .

Answer: $y = 4$ . Quadratic coefficients: $\frac{4}{1} = 4$ .

Flashcard 13: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 14: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$ ?

Answer: $y = 2$ . Simplify to $\frac{2x^2 + 4x}{x^2} = 2 + \frac{4}{x}$ .

Flashcard 15: Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$ .

Answer: $y = \frac{7}{3}$ . Cubic terms: $\frac{7}{3}$ .

Flashcard 16: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$ .

Answer: $y = \frac{3}{4}$ . Quadratic coefficients: $\frac{3}{4}$ .

Flashcard 17: Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$ .

Answer: $y = 4$ . Cubic terms give $\frac{4}{1} = 4$ .

Flashcard 18: Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$ .

Answer: $y = 0$ . Numerator degree less than denominator degree.

Flashcard 19: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$ .

Answer: $y = \frac{1}{2}$ . Quadratic coefficients: $\frac{1}{2}$ .

Flashcard 20: State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$ .

Answer: $y = 1$ . Equal-degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 21: What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 22: State the horizontal asymptote for $f(x) = \frac{3x - 4}{x + 5}$ .

Answer: $y = 3$ . Linear terms give ratio $\frac{3}{1} = 3$ .

Flashcard 23: What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?

Answer: $4$ . Leading coefficient ratio when degrees match.

Flashcard 24: Identify the horizontal asymptote of $f(x) = \frac{x^2}{x^2 - x + 1}$ .

Answer: $y = 1$ . Same degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 25: What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 26: What is the horizontal asymptote of $f(x) = \frac{2x^2 + x}{4x^2 + 3}$ ?

Answer: $y = \frac{1}{2}$ . Quadratic coefficients: $\frac{2}{4} = \frac{1}{2}$ .

Flashcard 27: What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?

Answer: $4$ . Leading coefficient ratio when degrees match.

Flashcard 28: What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 29: Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$ .

Answer: $y = 0$ . Numerator degree less than denominator degree.

Flashcard 30: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$ .

Answer: $y = 3$ . Quadratic terms: $\frac{3}{1} = 3$ .

Flashcard 31: Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$ .

Answer: $y = 4$ . Cubic terms give $\frac{4}{1} = 4$ .

Flashcard 32: What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?

Answer: $0$ . Numerator degree less than denominator degree.

Flashcard 33: Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$ .

Answer: $y = \frac{7}{3}$ . Cubic terms: $\frac{7}{3}$ .

Flashcard 34: Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$ .

Answer: $y = \frac{3}{2}$ . Fourth-degree leading coefficients: $\frac{3}{2}$ .

Flashcard 35: State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$ .

Answer: $y = 0$ . Linear over cubic approaches 0.

Flashcard 36: State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$ .

Answer: $y = \frac{1}{2}$ . Leading coefficient ratio: $\frac{1}{2}$ .

Flashcard 37: What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 38: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 39: Identify the horizontal asymptote of $f(x) = \frac{3x^2}{x^2 + 8}$ .

Answer: $y = 3$ . Same-degree quadratics: $\frac{3}{1} = 3$ .

Flashcard 40: What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 41: State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$ .

Answer: $y = 0$ . Linear over cubic approaches 0.

Flashcard 42: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$ .

Answer: $y = \frac{1}{2}$ . Quadratic coefficients: $\frac{1}{2}$ .

Flashcard 43: What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$ ?

Answer: $y = \frac{1}{3}$ . Quadratic leading coefficients: $\frac{1}{3}$ .

Flashcard 44: What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$ ?

Answer: $y = \frac{1}{4}$ . Cubic coefficients: $\frac{1}{4}$ .

Flashcard 45: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$ ?

Answer: $y = 2$ . Simplify to $\frac{2x^2 + 4x}{x^2} = 2 + \frac{4}{x}$ .

Flashcard 46: Identify the horizontal asymptote of $f(x) = \frac{3x^2}{x^2 + 8}$ .

Answer: $y = 3$ . Same-degree quadratics: $\frac{3}{1} = 3$ .

Flashcard 47: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$ ?

Answer: $y = \frac{5}{2}$ . Ratio of leading coefficients for equal-degree polynomials.

Flashcard 48: State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$ .

Answer: $y = 1$ . Equal-degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 49: State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$ .

Answer: $y = \frac{1}{2}$ . Leading coefficient ratio: $\frac{1}{2}$ .

Flashcard 50: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$ .

Answer: $y = \frac{3}{4}$ . Quadratic coefficients: $\frac{3}{4}$ .

Flashcard 51: What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 52: Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$ .

Answer: $y = \frac{7}{3}$ . Equal degree cubic polynomials: $\frac{7}{3}$ .

Flashcard 53: What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$ ?

Answer: $y = 0$ . Numerator degree less than denominator degree.

Flashcard 54: State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$ .

Answer: $y = \frac{1}{3}$ . Cubic leading coefficients: $\frac{1}{3}$ .

Flashcard 55: State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$ .

Answer: $y = \frac{1}{3}$ . Cubic leading coefficients: $\frac{1}{3}$ .

Flashcard 56: What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$ ?

Answer: $y = \frac{1}{4}$ . Cubic coefficients: $\frac{1}{4}$ .

Flashcard 57: State the horizontal asymptote for $f(x) = \frac{x^2 + 2x}{x^2 - x}$ .

Answer: $y = 1$ . Equal-degree quadratics: $\frac{1}{1} = 1$ .

Flashcard 58: Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$ .

Answer: $y = 1$ . Cubic terms: $\frac{1}{1} = 1$ .

Flashcard 59: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$ .

Answer: $y = 3$ . Quadratic terms: $\frac{3}{1} = 3$ .

Flashcard 60: Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$ .

Answer: $y = \frac{3}{2}$ . Fourth-degree leading coefficients: $\frac{3}{2}$ .

Flashcard 61: Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$ .

Answer: $y = 4$ . Quadratic coefficients: $\frac{4}{1} = 4$ .

Flashcard 62: What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?

Answer: $0$ . Numerator degree less than denominator degree.

Flashcard 63: What is the limit of $f(x) = \frac{3x^2}{2x^2 + 1}$ as $x$ approaches infinity?

Answer: $\frac{3}{2}$ . Divide leading coefficients when degrees are equal.

Flashcard 64: What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$ ?

Answer: $y = \frac{1}{3}$ . Quadratic leading coefficients: $\frac{1}{3}$ .

Flashcard 65: Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$ .

Answer: $y = 2$ . Fourth-degree terms dominate: $\frac{2}{1} = 2$ .

Flashcard 66: Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$ .

Answer: $y = \frac{7}{3}$ . Equal degree cubic polynomials: $\frac{7}{3}$ .

Flashcard 67: What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$ ?

Answer: None. Numerator degree exceeds denominator degree.

Flashcard 68: What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$ ?

Answer: $y = \frac{2}{5}$ . Quadratic terms: $\frac{2}{5}$ .

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$ ?

Answer: $y = \frac{2}{5}$ . Quadratic terms: $\frac{2}{5}$ .

Flashcard 70: What is the limit of $f(x) = \frac{3x^2}{2x^2 + 1}$ as $x$ approaches infinity?

Answer: $\frac{3}{2}$ . Divide leading coefficients when degrees are equal.

Flashcard 71: What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$ ?

Answer: $y = 0$ . Linear over quadratic approaches 0.

Flashcard 72: Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$ .

Answer: $y = 1$ . Cubic terms: $\frac{1}{1} = 1$ .

AP Calculus BC Flashcards: Limits At Infinity And Horizontal Asymptotes

Limits At Infinity And Horizontal Asymptotes

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How to use these flashcards

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Flashcard 1: Find the horizontal asymptote for f(x)=2x4+3xx4+x2f(x) = \frac{2x^4 + 3x}{x^4 + x^2}f(x)=x4+x22x4+3x​.

Flashcard 2: What is the horizontal asymptote of f(x)=2x2+x4x2+3f(x) = \frac{2x^2 + x}{4x^2 + 3}f(x)=4x2+32x2+x​?

Flashcard 3: State the horizontal asymptote for f(x)=3x−4x+5f(x) = \frac{3x - 4}{x + 5}f(x)=x+53x−4​.

Flashcard 4: Identify the horizontal asymptote of f(x)=x2x2−x+1f(x) = \frac{x^2}{x^2 - x + 1}f(x)=x2−x+1x2​.

Flashcard 5: What is the horizontal asymptote of f(x)=2x3x2+xf(x) = \frac{2x^3}{x^2 + x}f(x)=x2+x2x3​?

Flashcard 6: What is the horizontal asymptote of f(x)=x3+54x4+1f(x) = \frac{x^3 + 5}{4x^4 + 1}f(x)=4x4+1x3+5​?

Flashcard 7: What is the horizontal asymptote of f(x)=5xx2+xf(x) = \frac{5x}{x^2 + x}f(x)=x2+x5x​?

Flashcard 8: State the horizontal asymptote for f(x)=x2+2xx2−xf(x) = \frac{x^2 + 2x}{x^2 - x}f(x)=x2−xx2+2x​.

Flashcard 9: Identify the horizontal asymptote of f(x)=xx3+1f(x) = \frac{x}{x^3 + 1}f(x)=x3+1x​.

Flashcard 10: Identify the horizontal asymptote of f(x)=xx3+1f(x) = \frac{x}{x^3 + 1}f(x)=x3+1x​.

Flashcard 11: What is the horizontal asymptote of f(x)=5x32x3+7f(x) = \frac{5x^3}{2x^3 + 7}f(x)=2x3+75x3​?

Flashcard 12: Find the horizontal asymptote for f(x)=4x2−xx2+xf(x) = \frac{4x^2 - x}{x^2 + x}f(x)=x2+x4x2−x​.

Flashcard 13: What is the horizontal asymptote of f(x)=5x32x2+xf(x) = \frac{5x^3}{2x^2 + x}f(x)=2x2+x5x3​?

Flashcard 14: What is the horizontal asymptote of f(x)=2x2+4xx2f(x) = \frac{2x^2 + 4x}{x^2}f(x)=x22x2+4x​?

Flashcard 15: Determine the horizontal asymptote of f(x)=7x33x3+2xf(x) = \frac{7x^3}{3x^3 + 2x}f(x)=3x3+2x7x3​.

Flashcard 16: Determine the horizontal asymptote of f(x)=3x2+54x2+7f(x) = \frac{3x^2 + 5}{4x^2 + 7}f(x)=4x2+73x2+5​.

Flashcard 17: Find the horizontal asymptote for f(x)=4x3+1x3−2f(x) = \frac{4x^3 + 1}{x^3 - 2}f(x)=x3−24x3+1​.

Flashcard 18: Determine the horizontal asymptote of f(x)=4xx3+1f(x) = \frac{4x}{x^3 + 1}f(x)=x3+14x​.

Flashcard 19: Identify the horizontal asymptote of f(x)=x2+32x2−1f(x) = \frac{x^2 + 3}{2x^2 - 1}f(x)=2x2−1x2+3​.

Flashcard 20: State the horizontal asymptote for f(x)=x2−4x2+5f(x) = \frac{x^2 - 4}{x^2 + 5}f(x)=x2+5x2−4​.

Flashcard 21: What is the horizontal asymptote of f(x)=2xx2+1f(x) = \frac{2x}{x^2 + 1}f(x)=x2+12x​?

Flashcard 22: State the horizontal asymptote for f(x)=3x−4x+5f(x) = \frac{3x - 4}{x + 5}f(x)=x+53x−4​.

Flashcard 23: What is the limit of f(x)=4x2−x+6x2+x−12f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}f(x)=x2+x−124x2−x+6​ as xxx approaches infinity?

Flashcard 24: Identify the horizontal asymptote of f(x)=x2x2−x+1f(x) = \frac{x^2}{x^2 - x + 1}f(x)=x2−x+1x2​.

Flashcard 25: What is the horizontal asymptote of f(x)=4xx2+1f(x) = \frac{4x}{x^2 + 1}f(x)=x2+14x​?

Flashcard 26: What is the horizontal asymptote of f(x)=2x2+x4x2+3f(x) = \frac{2x^2 + x}{4x^2 + 3}f(x)=4x2+32x2+x​?

Flashcard 27: What is the limit of f(x)=4x2−x+6x2+x−12f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}f(x)=x2+x−124x2−x+6​ as xxx approaches infinity?

Flashcard 28: What is the horizontal asymptote of f(x)=2xx2+1f(x) = \frac{2x}{x^2 + 1}f(x)=x2+12x​?

Flashcard 29: Determine the horizontal asymptote of f(x)=4xx3+1f(x) = \frac{4x}{x^3 + 1}f(x)=x3+14x​.

Flashcard 30: Determine the horizontal asymptote of f(x)=3x2+2x2−xf(x) = \frac{3x^2 + 2}{x^2 - x}f(x)=x2−x3x2+2​.

Flashcard 31: Find the horizontal asymptote for f(x)=4x3+1x3−2f(x) = \frac{4x^3 + 1}{x^3 - 2}f(x)=x3−24x3+1​.

Flashcard 32: What is the limit of f(x)=6x33x4+2f(x) = \frac{6x^3}{3x^4 + 2}f(x)=3x4+26x3​ as xxx approaches infinity?

Flashcard 33: Determine the horizontal asymptote of f(x)=7x33x3+2xf(x) = \frac{7x^3}{3x^3 + 2x}f(x)=3x3+2x7x3​.

Flashcard 34: Find the horizontal asymptote for f(x)=3x4−2x2x4+5f(x) = \frac{3x^4 - 2x}{2x^4 + 5}f(x)=2x4+53x4−2x​.

Flashcard 35: State the horizontal asymptote for f(x)=2xx3+5f(x) = \frac{2x}{x^3 + 5}f(x)=x3+52x​.

Flashcard 36: State the horizontal asymptote of f(x)=x2+22x2−3f(x) = \frac{x^2 + 2}{2x^2 - 3}f(x)=2x2−3x2+2​.

Flashcard 37: What is the horizontal asymptote of f(x)=5xx2+xf(x) = \frac{5x}{x^2 + x}f(x)=x2+x5x​?

Flashcard 38: What is the horizontal asymptote of f(x)=5x32x2+xf(x) = \frac{5x^3}{2x^2 + x}f(x)=2x2+x5x3​?

Flashcard 39: Identify the horizontal asymptote of f(x)=3x2x2+8f(x) = \frac{3x^2}{x^2 + 8}f(x)=x2+83x2​.

Flashcard 40: What is the horizontal asymptote of f(x)=4x4x3+2f(x) = \frac{4x^4}{x^3 + 2}f(x)=x3+24x4​?

Flashcard 41: State the horizontal asymptote for f(x)=2xx3+5f(x) = \frac{2x}{x^3 + 5}f(x)=x3+52x​.

Flashcard 42: Identify the horizontal asymptote of f(x)=x2+32x2−1f(x) = \frac{x^2 + 3}{2x^2 - 1}f(x)=2x2−1x2+3​.

Flashcard 43: What is the horizontal asymptote of f(x)=x2+7x3x2−9f(x) = \frac{x^2 + 7x}{3x^2 - 9}f(x)=3x2−9x2+7x​?

Flashcard 44: What is the horizontal asymptote of f(x)=x3+24x3+xf(x) = \frac{x^3 + 2}{4x^3 + x}f(x)=4x3+xx3+2​?

Flashcard 45: What is the horizontal asymptote of f(x)=2x2+4xx2f(x) = \frac{2x^2 + 4x}{x^2}f(x)=x22x2+4x​?

Flashcard 46: Identify the horizontal asymptote of f(x)=3x2x2+8f(x) = \frac{3x^2}{x^2 + 8}f(x)=x2+83x2​.

Flashcard 47: What is the horizontal asymptote of f(x)=5x32x3+7f(x) = \frac{5x^3}{2x^3 + 7}f(x)=2x3+75x3​?

Flashcard 48: State the horizontal asymptote for f(x)=x2−4x2+5f(x) = \frac{x^2 - 4}{x^2 + 5}f(x)=x2+5x2−4​.

Flashcard 49: State the horizontal asymptote of f(x)=x2+22x2−3f(x) = \frac{x^2 + 2}{2x^2 - 3}f(x)=2x2−3x2+2​.

Flashcard 50: Determine the horizontal asymptote of f(x)=3x2+54x2+7f(x) = \frac{3x^2 + 5}{4x^2 + 7}f(x)=4x2+73x2+5​.

Flashcard 51: What is the horizontal asymptote of f(x)=2x3x2+xf(x) = \frac{2x^3}{x^2 + x}f(x)=x2+x2x3​?

Flashcard 52: Identify the horizontal asymptote of f(x)=7x3+53x3−2f(x) = \frac{7x^3 + 5}{3x^3 - 2}f(x)=3x3−27x3+5​.

Flashcard 53: What is the horizontal asymptote of f(x)=x3+54x4+1f(x) = \frac{x^3 + 5}{4x^4 + 1}f(x)=4x4+1x3+5​?

Flashcard 54: State the horizontal asymptote for f(x)=x3−x3x3+5f(x) = \frac{x^3 - x}{3x^3 + 5}f(x)=3x3+5x3−x​.

Flashcard 55: State the horizontal asymptote for f(x)=x3−x3x3+5f(x) = \frac{x^3 - x}{3x^3 + 5}f(x)=3x3+5x3−x​.

Flashcard 56: What is the horizontal asymptote of f(x)=x3+24x3+xf(x) = \frac{x^3 + 2}{4x^3 + x}f(x)=4x3+xx3+2​?

Flashcard 57: State the horizontal asymptote for f(x)=x2+2xx2−xf(x) = \frac{x^2 + 2x}{x^2 - x}f(x)=x2−xx2+2x​.

Flashcard 58: Find the horizontal asymptote for f(x)=x3x3+xf(x) = \frac{x^3}{x^3 + x}f(x)=x3+xx3​.

Flashcard 59: Determine the horizontal asymptote of f(x)=3x2+2x2−xf(x) = \frac{3x^2 + 2}{x^2 - x}f(x)=x2−x3x2+2​.

Flashcard 60: Find the horizontal asymptote for f(x)=3x4−2x2x4+5f(x) = \frac{3x^4 - 2x}{2x^4 + 5}f(x)=2x4+53x4−2x​.

Flashcard 61: Find the horizontal asymptote for f(x)=4x2−xx2+xf(x) = \frac{4x^2 - x}{x^2 + x}f(x)=x2+x4x2−x​.

Flashcard 62: What is the limit of f(x)=6x33x4+2f(x) = \frac{6x^3}{3x^4 + 2}f(x)=3x4+26x3​ as xxx approaches infinity?

Flashcard 63: What is the limit of f(x)=3x22x2+1f(x) = \frac{3x^2}{2x^2 + 1}f(x)=2x2+13x2​ as xxx approaches infinity?

Flashcard 64: What is the horizontal asymptote of f(x)=x2+7x3x2−9f(x) = \frac{x^2 + 7x}{3x^2 - 9}f(x)=3x2−9x2+7x​?

Flashcard 65: Find the horizontal asymptote for f(x)=2x4+3xx4+x2f(x) = \frac{2x^4 + 3x}{x^4 + x^2}f(x)=x4+x22x4+3x​.

Flashcard 66: Identify the horizontal asymptote of f(x)=7x3+53x3−2f(x) = \frac{7x^3 + 5}{3x^3 - 2}f(x)=3x3−27x3+5​.

Flashcard 67: What is the horizontal asymptote of f(x)=4x4x3+2f(x) = \frac{4x^4}{x^3 + 2}f(x)=x3+24x4​?

Flashcard 68: What is the horizontal asymptote of f(x)=2x2−5x5x2+xf(x) = \frac{2x^2 - 5x}{5x^2 + x}f(x)=5x2+x2x2−5x​?

Flashcard 69: What is the horizontal asymptote of f(x)=2x2−5x5x2+xf(x) = \frac{2x^2 - 5x}{5x^2 + x}f(x)=5x2+x2x2−5x​?

Flashcard 70: What is the limit of f(x)=3x22x2+1f(x) = \frac{3x^2}{2x^2 + 1}f(x)=2x2+13x2​ as xxx approaches infinity?

Flashcard 71: What is the horizontal asymptote of f(x)=4xx2+1f(x) = \frac{4x}{x^2 + 1}f(x)=x2+14x​?

Flashcard 72: Find the horizontal asymptote for f(x)=x3x3+xf(x) = \frac{x^3}{x^3 + x}f(x)=x3+xx3​.

Flashcard 1: Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$ .

Flashcard 2: What is the horizontal asymptote of $f(x) = \frac{2x^2 + x}{4x^2 + 3}$ ?

Flashcard 3: State the horizontal asymptote for $f(x) = \frac{3x - 4}{x + 5}$ .

Flashcard 4: Identify the horizontal asymptote of $f(x) = \frac{x^2}{x^2 - x + 1}$ .

Flashcard 5: What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$ ?

Flashcard 6: What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$ ?

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$ ?

Flashcard 8: State the horizontal asymptote for $f(x) = \frac{x^2 + 2x}{x^2 - x}$ .

Flashcard 9: Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$ .

Flashcard 10: Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$ .

Flashcard 11: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$ ?

Flashcard 12: Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$ .

Flashcard 13: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$ ?

Flashcard 14: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$ ?

Flashcard 15: Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$ .

Flashcard 16: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$ .

Flashcard 17: Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$ .

Flashcard 18: Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$ .

Flashcard 19: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$ .

Flashcard 20: State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$ .

Flashcard 21: What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$ ?

Flashcard 22: State the horizontal asymptote for $f(x) = \frac{3x - 4}{x + 5}$ .

Flashcard 23: What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?

Flashcard 24: Identify the horizontal asymptote of $f(x) = \frac{x^2}{x^2 - x + 1}$ .

Flashcard 25: What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$ ?

Flashcard 26: What is the horizontal asymptote of $f(x) = \frac{2x^2 + x}{4x^2 + 3}$ ?

Flashcard 27: What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?

Flashcard 28: What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$ ?

Flashcard 29: Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$ .

Flashcard 30: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$ .

Flashcard 31: Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$ .

Flashcard 32: What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?

Flashcard 33: Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$ .

Flashcard 34: Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$ .

Flashcard 35: State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$ .

Flashcard 36: State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$ .

Flashcard 37: What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$ ?

Flashcard 38: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$ ?

Flashcard 39: Identify the horizontal asymptote of $f(x) = \frac{3x^2}{x^2 + 8}$ .

Flashcard 40: What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$ ?

Flashcard 41: State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$ .

Flashcard 42: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$ .

Flashcard 43: What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$ ?

Flashcard 44: What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$ ?

Flashcard 45: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$ ?

Flashcard 46: Identify the horizontal asymptote of $f(x) = \frac{3x^2}{x^2 + 8}$ .

Flashcard 47: What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$ ?

Flashcard 48: State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$ .

Flashcard 49: State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$ .

Flashcard 50: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$ .

Flashcard 51: What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$ ?

Flashcard 52: Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$ .

Flashcard 53: What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$ ?

Flashcard 54: State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$ .

Flashcard 55: State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$ .

Flashcard 56: What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$ ?

Flashcard 57: State the horizontal asymptote for $f(x) = \frac{x^2 + 2x}{x^2 - x}$ .

Flashcard 58: Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$ .

Flashcard 59: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$ .

Flashcard 60: Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$ .

Flashcard 61: Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$ .

Flashcard 62: What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?

Flashcard 63: What is the limit of $f(x) = \frac{3x^2}{2x^2 + 1}$ as $x$ approaches infinity?

Flashcard 64: What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$ ?

Flashcard 65: Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$ .

Flashcard 66: Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$ .

Flashcard 67: What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$ ?

Flashcard 68: What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$ ?

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$ ?

Flashcard 70: What is the limit of $f(x) = \frac{3x^2}{2x^2 + 1}$ as $x$ approaches infinity?

Flashcard 71: What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$ ?

Flashcard 72: Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$ .